Role of mass asymmetry in fusion of super-heavy nuclei. K. Siwek- Wilczyńska, I. Skwira-Chalot, J. Wilczyński. aim to compare our model predictions with the measured (Dubna , GSI , Riken) evaporation -residue cross sections for synthesis of super-heavy nuclei in xn reactions . - PowerPoint PPT Presentation
Role of mass asymmetry in fusion of super-heavy nuclei K. Siwek- Wilczyńska, I. Skwira-Chalot, J. Wilczyński aim to compare our model predictions with the measured (Dubna, GSI, Riken) evaporation-residue cross sections for synthesis of super-heavy nuclei in xn reactions. to predict cross sections for the synthesis of new super-heavy nuclei in cold and hot fusion reactions. to verify the fusion hindrance factor (strongly dependent on the mass asymmetry).
Transcript
Role of mass asymmetry in fusion of super-heavy nuclei
K. Siwek- Wilczyńska, I. Skwira-Chalot, J. Wilczyński
aim to compare our model predictions with the measured (Dubna, GSI, Riken) evaporation-residue cross sections for synthesis of super-heavy nuclei in xn reactions.
to predict cross sections for the synthesis of new super-heavy nuclei in cold and hot fusion reactions.
to verify the fusion hindrance factor (strongly dependent on the mass asymmetry).
(synthesis) = (capture) × P(fusion) × P(survive)
Systematic studies based on experimental data:
(capture) - the „diffused-barrier formula” ( 3 parameters):
W. Świątecki, K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 71 (2005) 014602,
Acta Phys. Pol. B34(2003) 2049
A 2 fit to 48 experimental near-barrier fusion excitation functions
in the range of 40 < ZCN < 98 resulted in systematics that allow
us to predict values of the three parameters B0, w, R
(K. Siwek-Wilczyńska, J. Wilczyński Phys. Rev. C 69 (2004) 024611)
2)exp()1()( 22
E
wXXerfXREcap
integral.errorGaussian,2
: 0 XerfwBE
Xwhere
Formula derived assuming:• Gaussian shape of the fusion barrier distribution• Classical expression for σfus(E,B)=πR2(1-B/E)
For very heavy systems a range of partial waves contributing to CN formation is limited (critical angular momenta for disappearing macroscopic fission barrier).
We propose:
cap(subcritical l ) =
cap for E ≤ Bo
cap
(subcritical l ) = cap
(E=Bo)*Bo/Ec.m.
for E > Bo
max
0*
max
2212
iE
iiii
iiii
i dE
Es
m
P(survive) – Statistical model (Monte Carlo method)
Partial widths for emission of light particles – Weisskopf formula
PVBEEE Cii
iroti *maxwhere:
The fission width (transition state method), E*< 40 MeV
max
0*
max
2
1 fEffiss
fiss dKE
KE
Upper limit of the final-state excitation energy after emission of a particle i
PsaddleEsaddleEE rotf )()(*max Upper limit of the thermal excitation energy at the saddle
i – cross section for the production of the compound nucleus in the inverse process
mi, si , εi - mass, spin and kinetic energy of the emitted particle
ρ, ρi – level densities of the parent and daughter nuclei
The level density is calculated using the Fermi-gas-model formula aEE 2exp
included as proposed by Ignatyuk(A.V. Ignatyuk et al., Sov. J. Nucl. Phys. 29 (1975) 255)
dEUshell
macro eU
aa 11
• Shell effects
where: U - excitation energy, Ed - damping parameter
shell – shell correction energy, δshell (g.s.) (Möller et al., At. Data Nucl. Data Tables 59 (1995) 185), δshell(saddle)≈ 0
MeVEd 5.18
jk
jsmacro BArBArAra 31
0322
03
0 1426.01355.004543.0 fmr 153.10
Bs , Bk ( W.D. Myers and W.J. Świątecki, Ann. Phys. 84 (1974) 186)
,
(W. Reisdorf, Z. Phys. A. – Atoms and Nuclei 300 (1981) 227)
Lines - calculations using Smoluchowski diffusion equation
P(fusion) = ½(1-erf√B/T) B = bx02/2
if x0 ≥ 0 (injection point),
Smoluchowski Diffusion Equation
),(),(),(
2
2
txWx
TtxbxWxt
txW
Viscosity of fluidDriving force
W(x,t) = probability to find Brownian particle at position x at time t Exact solution in a parabolic potential
V(x) = -bx2/2 is a sliding, swelling Gaussian.
P(fusion) = fraction of Gaussian captured inside the barrier as t→∞
Temperature
injection point
saddle
W.J. Świątecki, K. Siwek-Wilczyńska, J. WilczyńskiActa Phys. Pol. B34 (2003)2049, IJMP E13 (2004) 261, Phys.Rev.C71 (2005) 014602
- Ch. Düllmann et al. Nature 418 (2002) 859,
A. Türler et al. Eur. Phys. J. A17 (2003) 505
Test of P(fusion) as a function of entrance channel asymmetry and excitation energy
Calculations for Z=108 (different mass asymmetries):
26Mg + 248
Cm →274-xnHs, 58
Fe+208
Pb →266-xn
Hs, 136Xe+
136Xe →
272-xnHs
S. Hofmann, Rep. Prog. Phys. 61 (1998) 639; and private communication
136Xe + 136Xe → 272Hsexperiment underway at Dubna
Summary:
1. SHE production cross sections (synthesis) = (capture) × P(fusion) × P(survive)
2. We have well tested tools for calculating capture cross sections
(capture) and statistical decay of very heavy nuclei P(survive).For P(fusion) we use a simple model based on the Smoluchowski diffusion equation. This model works quite well for cold fusion reactions.
3. It is essential to test predictions of P(fusion) for hot fusion reactions and more symmetric systems which probably will be used in attempts to synthesize new SHE (Z=120 and beyond), therefore
comparison of Mg+Cm, Fe+Pb and Xe+Xe reactions is very important.
